Base change for Quot-scheme

Question

I am reading the book of Huybrechts and Lehn "The Geometry of Moduli Spaces of Sheaves" with an aim to become a little bit familiar with this topic. Now I am trying to understand what is Grothendieck's Quot-scheme and solve the following very basic question:

Let $f:X\rightarrow S$ be a projective morphism of schemes of finite type, $P\in\mathbb{Q}[z]$ be a fixed polynomial, $\mathcal{H}$ be a coherent sheaf on $X$ and $\text{Quot}_{X/S}(\mathcal{H}, P)\xrightarrow{\pi}S$ be a Quot-scheme for the morphism $f$, the sheaf $\mathcal{H}$ and Hilbert polynomial $P$. Let $\varphi:T\rightarrow S$ be the arbitrary morphism of schemes, $X_T:=X\times_ST$, $\varphi_X:X_T\rightarrow X$ and $f_T:X_T\rightarrow T$, $\mathcal{H}_T:=\varphi^*_X\mathcal{H}$. Then there exists isomorphism of schemes

$$\text{Quot}_{X_T/T}(\mathcal{H}_T, P)\cong T\times_S\text{Quot}_{X/S}(\mathcal{H}, P).$$

My guess is that the required isomorphism is just the map $$\xi:\text{Quot}_{X_T/T}(\mathcal{H}_T, P)\rightarrow T\times_S\text{Quot}_{X/S}(\mathcal{H}, P),$$

which is obtained from the universal property of fiber product. However I can't prove that it is invertible.

Can anyone help me with this question or give a suitable reference?

Answer 1

The idea is to consider the two sides $\text{Quot}_{X_T/T}(\mathcal{H}_T, P)$ and $ T\times_S\text{Quot}_{X/S}(\mathcal{H}, P)$ as $T$-schemes, to show that their functors of points are isomorphic and to apply Yoneda. In fact, one does not even need to know that the functors are actually representable, and the argument works even in situations when they are not (quasi-projective, or proper non-projective).

Let $g:V\rightarrow T\in \text{Sch}/T$ . We have by definition $$ \text{Quot}_{X_T/T}(\mathcal{H}_T, P)(V)=\{g^*\mathcal{H}_T\twoheadrightarrow \mathcal{F} |\ \mathcal{F}\text{ coherent sheaf on } X_T\times_T V, \text{flat over V , with }\forall t\in T,\ \text{Hilb}_{X_t}(\mathcal{F}_t)=P\} $$ and using the property of the fiber product and the fact that we are looking at the functor on $T$-schemes, with a fixed morphism to $T$, we have $$ T\times_S \text{Quot}_{X/S}(\mathcal{H}, P)(V)=\{(\phi\circ g)^*\mathcal{H}\twoheadrightarrow \mathcal{G} |\ \mathcal{G}\text{ coherent sheaf on $X\times_S V$, flat over $V$, with }\forall s\in S,\ \text{Hilb}_{X_s}(\mathcal{G}_s)=P\} $$ To compare the two, we then use

• The canonical isomorphism $i:X\times_S V \simeq X_T\times_T V$
• The isomorphism of functors on categories of coherent sheaves $\eta:(\phi\circ g)^*\simeq g^*\phi^*$
• The fact that the $i^*$ and $\eta$ are compatible in the obvious way, by coherence of isomorphisms of type $\eta$.
• The fact that the Hilbert polynomials (which are computed with respect to compatible line bundles coming from $S$) of $\mathcal{F}_t$ and $(i^*\mathcal{F})_{f(t)}$ coincide.